Introduction to Fuzzy Logic Control Andrew L. Nelson Visiting Research Faculty University of South Florida

Overview Outline to the left in green Current topic in yellow References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Outline to the left in green Current topic in yellow References Introduction Crisp Variables Fuzzy Variables Fuzzy Logic Operators Fuzzy Control Case Study 2/9/2004 Fuzzy Logic

References References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary L. Zadah, “Fuzzy sets as a basis of possibility” Fuzzy Sets Systems, Vol. 1, pp3-28, 1978. T. J. Ross, “Fuzzy Logic with Engineering Applications”, McGraw-Hill, 1995. K. M. Passino, S. Yurkovich, "Fuzzy Control" Addison Wesley, 1998. 2/9/2004 Fuzzy Logic

Introduction Fuzzy logic: References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Fuzzy logic: A way to represent variation or imprecision in logic A way to make use of natural language in logic Approximate reasoning Humans say things like "If it is sunny and warm today, I will drive fast" Linguistic variables: Temp: {freezing, cool, warm, hot} Cloud Cover: {overcast, partly cloudy, sunny} Speed: {slow, fast} 2/9/2004 Fuzzy Logic

Crisp (Traditional) Variables Crisp variables represent precise quantities: x = 3.1415296 A {0,1} A proposition is either True or False A  B  C King(Richard)  Greedy(Richard)  Evil(Richard) Richard is either greedy or he isn't: Greedy(Richard) {0,1} References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary 2/9/2004 Fuzzy Logic

Fuzzy Sets What if Richard is only somewhat greedy? Fuzzy Sets can represent the degree to which a quality is possessed. Fuzzy Sets (Simple Fuzzy Variables) have values in the range of [0,1] Greedy(Richard) = 0.7 Question: How evil is Richard? References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary 2/9/2004 Fuzzy Logic

Fuzzy Linguistic Variables Fuzzy Linguistic Variables are used to represent qualities spanning a particular spectrum Temp: {Freezing, Cool, Warm, Hot} Membership Function Question: What is the temperature? Answer: It is warm. Question: How warm is it? References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary 2/9/2004 Fuzzy Logic

Membership Functions Temp: {Freezing, Cool, Warm, Hot} References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Temp: {Freezing, Cool, Warm, Hot} Degree of Truth or "Membership" 2/9/2004 Fuzzy Logic

Membership Functions How cool is 36 F° ? References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary How cool is 36 F° ? 2/9/2004 Fuzzy Logic

Membership Functions How cool is 36 F° ? References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary How cool is 36 F° ? It is 30% Cool and 70% Freezing 0.7 0.3 2/9/2004 Fuzzy Logic

Fuzzy Logic References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary How do we use fuzzy membership functions in predicate logic? Fuzzy logic Connectives: Fuzzy Conjunction,  Fuzzy Disjunction,  Operate on degrees of membership in fuzzy sets 2/9/2004 Fuzzy Logic

Fuzzy Disjunction AB max(A, B) References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary AB max(A, B) AB = C "Quality C is the disjunction of Quality A and B" (AB = C)  (C = 0.75) 2/9/2004 Fuzzy Logic

Fuzzy Conjunction AB min(A, B) References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary AB min(A, B) AB = C "Quality C is the conjunction of Quality A and B" (AB = C)  (C = 0.375) 2/9/2004 Fuzzy Logic

Example: Fuzzy Conjunction Calculate AB given that A is .4 and B is 20 References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary 2/9/2004 Fuzzy Logic

Example: Fuzzy Conjunction Calculate AB given that A is .4 and B is 20 References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Determine degrees of membership: 2/9/2004 Fuzzy Logic

Example: Fuzzy Conjunction Calculate AB given that A is .4 and B is 20 References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary 0.7 Determine degrees of membership: A = 0.7 2/9/2004 Fuzzy Logic

Example: Fuzzy Conjunction Calculate AB given that A is .4 and B is 20 References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary 0.9 0.7 Determine degrees of membership: A = 0.7 B = 0.9 2/9/2004 Fuzzy Logic

Example: Fuzzy Conjunction Calculate AB given that A is .4 and B is 20 References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary 0.9 0.7 Determine degrees of membership: A = 0.7 B = 0.9 Apply Fuzzy AND AB = min(A, B) = 0.7 2/9/2004 Fuzzy Logic

Fuzzy Control References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Fuzzy Control combines the use of fuzzy linguistic variables with fuzzy logic Example: Speed Control How fast am I going to drive today? It depends on the weather. Disjunction of Conjunctions 2/9/2004 Fuzzy Logic

Inputs: Temperature Temp: {Freezing, Cool, Warm, Hot} References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Temp: {Freezing, Cool, Warm, Hot} 2/9/2004 Fuzzy Logic

Inputs: Temperature, Cloud Cover References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Temp: {Freezing, Cool, Warm, Hot} Cover: {Sunny, Partly, Overcast} 2/9/2004 Fuzzy Logic

Output: Speed Speed: {Slow, Fast} References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Speed: {Slow, Fast} 2/9/2004 Fuzzy Logic

Rules If it's Sunny and Warm, drive Fast References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary If it's Sunny and Warm, drive Fast Sunny(Cover)Warm(Temp) Fast(Speed) If it's Cloudy and Cool, drive Slow Cloudy(Cover)Cool(Temp) Slow(Speed) Driving Speed is the combination of output of these rules... 2/9/2004 Fuzzy Logic

Example Speed Calculation References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary How fast will I go if it is 65 F° 25 % Cloud Cover ? 2/9/2004 Fuzzy Logic

Fuzzification: Calculate Input Membership Levels References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary 65 F°  Cool = 0.4, Warm= 0.7 2/9/2004 Fuzzy Logic

Fuzzification: Calculate Input Membership Levels References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary 65 F°  Cool = 0.4, Warm= 0.7 25% Cover Sunny = 0.8, Cloudy = 0.2 2/9/2004 Fuzzy Logic

...Calculating... If it's Sunny and Warm, drive Fast Sunny(Cover)Warm(Temp)Fast(Speed) 0.8  0.7 = 0.7  Fast = 0.7 If it's Cloudy and Cool, drive Slow Cloudy(Cover)Cool(Temp)Slow(Speed) 0.2  0.4 = 0.2  Slow = 0.2 References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary 2/9/2004 Fuzzy Logic

Defuzzification: Constructing the Output References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Speed is 20% Slow and 70% Fast Find centroids: Location where membership is 100% 2/9/2004 Fuzzy Logic

Defuzzification: Constructing the Output References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Speed is 20% Slow and 70% Fast Find centroids: Location where membership is 100% 2/9/2004 Fuzzy Logic

Defuzzification: Constructing the Output References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Speed is 20% Slow and 70% Fast Speed = weighted mean = (2*25+... 2/9/2004 Fuzzy Logic

Defuzzification: Constructing the Output References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Speed is 20% Slow and 70% Fast Speed = weighted mean = (2*25+7*75)/(9) = 63.8 mph 2/9/2004 Fuzzy Logic

Notes: Follow-up Points References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Fuzzy Logic Control allows for the smooth interpolation between variable centroids with relatively few rules This does not work with crisp (traditional Boolean) logic Provides a natural way to model some types of human expertise in a computer program 2/9/2004 Fuzzy Logic

Notes: Drawbacks to Fuzzy logic References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Requires tuning of membership functions Fuzzy Logic control may not scale well to large or complex problems Deals with imprecision, and vagueness, but not uncertainty 2/9/2004 Fuzzy Logic

Summary References Introduction Crisp Variables Fuzzy Sets Linguistic Variables Membership Functions Fuzzy Logic Fuzzy OR Fuzzy AND Example Fuzzy Control Variables Rules Fuzzification Defuzzification Summary Fuzzy Logic provides way to calculate with imprecision and vagueness Fuzzy Logic can be used to represent some kinds of human expertise Fuzzy Membership Sets Fuzzy Linguistic Variables Fuzzy AND and OR Fuzzy Control 2/9/2004 Fuzzy Logic